AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Average Error: 26.5 → 7.7

Time: 6.0s

Precision: 64

Math

\[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} = -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 3.8381343719672391 \cdot 10^{237}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r812777 = x;
        double r812778 = y;
        double r812779 = r812777 + r812778;
        double r812780 = z;
        double r812781 = r812779 * r812780;
        double r812782 = t;
        double r812783 = r812782 + r812778;
        double r812784 = a;
        double r812785 = r812783 * r812784;
        double r812786 = r812781 + r812785;
        double r812787 = b;
        double r812788 = r812778 * r812787;
        double r812789 = r812786 - r812788;
        double r812790 = r812777 + r812782;
        double r812791 = r812790 + r812778;
        double r812792 = r812789 / r812791;
        return r812792;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r812793 = x;
        double r812794 = y;
        double r812795 = r812793 + r812794;
        double r812796 = z;
        double r812797 = r812795 * r812796;
        double r812798 = t;
        double r812799 = r812798 + r812794;
        double r812800 = a;
        double r812801 = r812799 * r812800;
        double r812802 = r812797 + r812801;
        double r812803 = b;
        double r812804 = r812794 * r812803;
        double r812805 = r812802 - r812804;
        double r812806 = r812793 + r812798;
        double r812807 = r812806 + r812794;
        double r812808 = r812805 / r812807;
        double r812809 = -inf.0;
        bool r812810 = r812808 <= r812809;
        double r812811 = 3.838134371967239e+237;
        bool r812812 = r812808 <= r812811;
        double r812813 = !r812812;
        bool r812814 = r812810 || r812813;
        double r812815 = r812800 + r812796;
        double r812816 = r812815 - r812803;
        double r812817 = 1.0;
        double r812818 = r812807 / r812805;
        double r812819 = r812817 / r812818;
        double r812820 = r812814 ? r812816 : r812819;
        return r812820;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	26.5
Target	11.2
Herbie	7.7

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt -3.5813117084150564 \cdot 10^{153}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \lt 1.2285964308315609 \cdot 10^{82}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < -inf.0 or 3.838134371967239e+237 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y))

   1. Initial program 61.4

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
   2. Using strategy rm

   3. Applied clear-num 61.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]

   4. Taylor expanded around 0 17.4

      \[\leadsto \color{blue}{\left(a + z\right) - b}\]

   if -inf.0 < (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) < 3.838134371967239e+237

   1. Initial program 0.3

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y}\]
   2. Using strategy rm

   3. Applied clear-num 0.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} = -\infty \lor \neg \left(\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \le 3.8381343719672391 \cdot 10^{237}\right):\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x + t\right) + y}{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))